A while ago Joel Hamkins suggested that I copy cat this post of Norman’s: Chart of large cardinals near high-jump with my own large cardinal chart. So here is the large cardinal chart of Ramsey-like cardinals and their relationship to other large cardinal notions. The Ramsey-like cardinals: $\alpha$-*iterable* cardinals ($1\leq\alpha\leq\omega_1$), *strongly Ramsey* cardinals, *super Ramsey* cardinals, and *superlatively Ramsey* cardinals were introduced by myself and/or Philip Welch. All relevant definitions are here.

### Recent Writing

- Virtual large cardinal principles
- Filter games and Ramsey-like cardinals
- The exact strength of the class forcing theorem
- A model of the generic Vopěnka principle in which the ordinals are not $\Delta_2$-Mahlo
- A model of second-order arithmetic with the choice scheme in which $\Pi^1_2$-dependent choice fails

### Mathoverflow Activity

- Comment by Victoria Gitman on Does Con(ZF + Reinhardt) really imply Con(ZFC + I0)?@Stefan For some reason, I cannot click on the link.Victoria Gitman
- Comment by Victoria Gitman on Cofinality of $j(\kappa)$ for a measurability embedding $j:V\to M$ with critical point $\kappa$@AsafKaragila I edited the question to make it clearer. Maybe I will ask your version as a follow-up if you don't ask it first :).Victoria Gitman
- Comment by Victoria Gitman on Cofinality of $j(\kappa)$ for a measurability embedding $j:V\to M$ with critical point $\kappa$@AsafKaragila I should have done a better job stating my question clearly. I wanted to know precisely what Yair answered: whether there are models where $\text{cf}(j(\kappa))$ is smaller than the size of $j(\kappa)$. Your interpretation of the question is very interesting as well, but I think much harder to answer.Victoria Gitman

- Comment by Victoria Gitman on Does Con(ZF + Reinhardt) really imply Con(ZFC + I0)?
### Cantor’s Attic

- Constructible universeImplications, equivalences, and consequences of $0^\#$'s existence ← Older revision Revision as of 11:26, 14 December 2017 Line 63: Line 63: If $0^\#$ exists then: If $0^\#$ exists then: −* $\aleph_\omega$ is [[stable]] in $L$ and so $0^\#$ also corresponds to the set of the Gödel numberings of first-order formulas $\varphi$ such that $L\models\varphi(\aleph_0,\aleph_1...\aleph_n)$+* $L_{\aleph_\omega}\prec […]Julian Barathieu
- Axiom of determinacy← Older revision Revision as of 11:13, 14 December 2017 Line 100: Line 100: * Every uncountable cardinal $Julian Barathieu
- Upper attic← Older revision Revision as of 11:07, 14 December 2017 Line 30: Line 30: * [[zero dagger| $0^\dagger$]], $j:L[U]\to L[U]$ cardinal * [[zero dagger| $0^\dagger$]], $j:L[U]\to L[U]$ cardinal * '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal * '''[[measurable]]''' cardinal, [[weakly measurable]] cardinal, singular [[Jonsson|Jónsson]] cardinal −* [[Jonsson | Jónsson]] cardinal, [[Rowbottom]] cardinal, '''[[Ramsey]]''' cardinal, [[strongly Ramsey]] […]Julian Barathieu
- Strongly compactDiverse characterizations ← Older revision Revision as of 08:57, 14 December 2017 Line 18: Line 18: A cardinal $\kappa$ is $\theta$-strongly compact if and only if there is an [[elementary embedding]] $j:V\to M$ of the set-theoretic universe $V$ into a transitive class $M$ with critical point $\kappa$, such that $j''\theta\subset s\in M$ for some […]Julian Barathieu
- Supercompact← Older revision Revision as of 21:12, 11 December 2017 Line 10: Line 10: One can see the equivalence of the two formulations by first considering the ultrafilter $U$ arising from the [[seed]] $j''\theta$, so that $X\in U\iff j''\theta\in j(X)$. It is easy to check that $U$ is a normal fine measure on $\mathcal{P}_\kappa(\theta)$. Conversely, […]Julian Barathieu

- Constructible universe

Yay! I love it!